Non-Abelian Braiding Protected by Quaternion Topological Charges

The concept of braiding groups in relation to elementary particles is an interesting topic in physics. Anyonic quasiparticles protected by topological order can host non-Abelian braiding characteristics. However, the realization of topological protected non-Abelian braiding remain challenging in quantum systems. Recent, non-Abelian topological phase has been proposed as a novel multiband topological phase and experimentally realized in microwaves and acoustics. Its band topology is classified by a non-Abelian group, the quaternion group Q₈={±i,±j,±k,-1,1}. The eight group elements obey the following relations as, ij=k,jk=i,ki=j and i²=j²=k²=-1. Here, we demonstrate that each non-Abelian quaternion charge gives rise to a unique topological non-Abelian gauge field, enabling the Bloch eigenmodes to exhibit braiding dynamics akin to anyons when driven across the Brillouin zone. We show that each non-Abelian quaternion charge corresponds to a unique topologically protected braiding sequence. An acoustic system with built-in synthetic dimensions is proposed for the realization of this exotic non-Abelian topological phase and illustrating the braiding process.